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Properties

Label 1-1441-1441.449-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.969 + 0.246i$
Analytic cond. $154.856$
Root an. cond. $154.856$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.644 + 0.764i)2-s + (−0.681 − 0.732i)3-s + (−0.168 − 0.985i)4-s + (−0.906 + 0.421i)5-s + (0.998 − 0.0483i)6-s + (−0.354 + 0.935i)7-s + (0.861 + 0.506i)8-s + (−0.0724 + 0.997i)9-s + (0.262 − 0.964i)10-s + (−0.607 + 0.794i)12-s + (−0.354 + 0.935i)13-s + (−0.485 − 0.873i)14-s + (0.926 + 0.377i)15-s + (−0.943 + 0.331i)16-s + (0.607 + 0.794i)17-s + (−0.715 − 0.698i)18-s + ⋯
L(s)  = 1  + (−0.644 + 0.764i)2-s + (−0.681 − 0.732i)3-s + (−0.168 − 0.985i)4-s + (−0.906 + 0.421i)5-s + (0.998 − 0.0483i)6-s + (−0.354 + 0.935i)7-s + (0.861 + 0.506i)8-s + (−0.0724 + 0.997i)9-s + (0.262 − 0.964i)10-s + (−0.607 + 0.794i)12-s + (−0.354 + 0.935i)13-s + (−0.485 − 0.873i)14-s + (0.926 + 0.377i)15-s + (−0.943 + 0.331i)16-s + (0.607 + 0.794i)17-s + (−0.715 − 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.969 + 0.246i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.969 + 0.246i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2002146033 + 0.02503176326i\)
\(L(\frac12)\) \(\approx\) \(0.2002146033 + 0.02503176326i\)
\(L(1)\) \(\approx\) \(0.3705638186 + 0.1470186205i\)
\(L(1)\) \(\approx\) \(0.3705638186 + 0.1470186205i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.644 + 0.764i)T \)
3 \( 1 + (-0.681 - 0.732i)T \)
5 \( 1 + (-0.906 + 0.421i)T \)
7 \( 1 + (-0.354 + 0.935i)T \)
13 \( 1 + (-0.354 + 0.935i)T \)
17 \( 1 + (0.607 + 0.794i)T \)
19 \( 1 + (-0.958 + 0.285i)T \)
23 \( 1 + (-0.995 - 0.0965i)T \)
29 \( 1 + (0.970 + 0.239i)T \)
31 \( 1 + (-0.399 - 0.916i)T \)
37 \( 1 + (-0.885 + 0.464i)T \)
41 \( 1 + (0.0241 - 0.999i)T \)
43 \( 1 + (-0.906 + 0.421i)T \)
47 \( 1 + (0.0724 - 0.997i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.607 - 0.794i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.906 + 0.421i)T \)
71 \( 1 + (-0.215 + 0.976i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.485 + 0.873i)T \)
83 \( 1 + (0.168 - 0.985i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.779 - 0.626i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.436524171011151798592412716, −19.81276770697411493455735345726, −19.27801230999801764623444840629, −18.10657930974700690447075754395, −17.5038855381679133555023007547, −16.75357986386927394983157027610, −16.171571255417658831015578873918, −15.62621306315302633228707652667, −14.50415144452747380872290568012, −13.37887675588871787390874054110, −12.4417093242572014527680162237, −12.05757731765254996640634608520, −11.12490863149691002965710970395, −10.47288173966134981903492959271, −9.943718841864383304368823479034, −9.04105192818782655234400836738, −8.13150820187781664737507258130, −7.399232200921322111924659800618, −6.484401593782967502145505802871, −5.06197345350593898384341081784, −4.42609694885279915508822003901, −3.60035893913076599987653932693, −2.96527413120341242244275004725, −1.2551501514890113289735421122, −0.36024563194998223825542248823, 0.15365613772160536258471361565, 1.598530923828921037121570125557, 2.39243318977121949105819136884, 3.917750830780701751015593422500, 4.94165243138055063217475635647, 5.89224196312699264710595677166, 6.53212684611673589347306426982, 7.10944548840285095229844993657, 8.20233789593678261868908765497, 8.44348682352654147414014232645, 9.751681584671494608573283754522, 10.52018063167207977367140938677, 11.39323401484958964009784126431, 12.09700218237100611884313367725, 12.72808899527113749530770443541, 13.97930893044952672678615946173, 14.62818051494552181577674009122, 15.47758191581073219585995518634, 16.11325717034432001003369466013, 16.839918936673503387388191264178, 17.49998745579907119836648705514, 18.578767969832411199851837217928, 18.819966686522099117426234940566, 19.32283620067349243887014892839, 20.145997242914251914160425466225

Graph of the $Z$-function along the critical line